L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.29 + 2.70i)3-s − 3i·4-s + (0.707 + 4.94i)5-s + (2.82 − i)6-s − 7·7-s + (−4.94 + 4.94i)8-s + (−5.65 − 7i)9-s + (3.00 − 4.00i)10-s + 9.89i·11-s + (8.12 + 3.87i)12-s + (−8 + 8i)13-s + (4.94 + 4.94i)14-s + (−14.3 − 4.48i)15-s − 4.99·16-s + (−18.3 + 18.3i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.353i)2-s + (−0.430 + 0.902i)3-s − 0.750i·4-s + (0.141 + 0.989i)5-s + (0.471 − 0.166i)6-s − 7-s + (−0.618 + 0.618i)8-s + (−0.628 − 0.777i)9-s + (0.300 − 0.400i)10-s + 0.899i·11-s + (0.676 + 0.323i)12-s + (−0.615 + 0.615i)13-s + (0.353 + 0.353i)14-s + (−0.954 − 0.299i)15-s − 0.312·16-s + (−1.08 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.199186 + 0.435903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.199186 + 0.435903i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.29 - 2.70i)T \) |
| 5 | \( 1 + (-0.707 - 4.94i)T \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + 4iT^{2} \) |
| 11 | \( 1 - 9.89iT - 121T^{2} \) |
| 13 | \( 1 + (8 - 8i)T - 169iT^{2} \) |
| 17 | \( 1 + (18.3 - 18.3i)T - 289iT^{2} \) |
| 19 | \( 1 - 10T + 361T^{2} \) |
| 23 | \( 1 + (-24.0 + 24.0i)T - 529iT^{2} \) |
| 29 | \( 1 - 4.24T + 841T^{2} \) |
| 31 | \( 1 + 14iT - 961T^{2} \) |
| 37 | \( 1 + (-30 + 30i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 33.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36 - 36i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (4.24 - 4.24i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (70.7 - 70.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 29.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-32 + 32i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (39 - 39i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 56iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (36.7 + 36.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 19.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (113 + 113i)T + 9.40e3iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.23987092596819511433947671530, −12.66937468176768360713119970749, −11.36684790647293359202816382298, −10.58129187467465833691244952591, −9.811745863501089222348002304897, −9.112816842924075043703766906012, −6.85351808989082253946203402855, −6.01383823634531880522213980619, −4.41699730211243890148507602821, −2.62513825992196627379721604839,
0.39930638394500475769208940704, 3.02843516296389045995324115271, 5.18706757672327632885740187913, 6.52546694774950296529132460344, 7.53534504763724249964030553505, 8.643648022220908877723773114105, 9.547207636078389798051550982679, 11.37290627385305544747522396540, 12.27013318410799981885729194963, 13.15277827553338078464316547772